Well-Structured Mathematical Logic Damon Scott Department of Mathematics Francis Marion University CAROLINA ACADEMIC PRESS Durham, North Carolina
Copyright 2013 Damon Scott All Rights Reserved Library of Congress Cataloging-in-Publication Data Scott, Damon, 1962 - Well-structured mathematical logic / Damon Scott. pages cm Includes bibliographical references and index. ISBN 978-1 - 61163-368 - 9 1. Logic, Symbolic and mathematical. I. Title. QA9.S4175 2012 511.3 dc23 2012041322 Carolina Academic Press 700 Kent Street Durham, North Carolina 27701 Telephone (919) 489-7486 Fax (919) 493-5668 www.cap-press.com Printed in the United States of America
Table of Contents Table of Contents.............................. v Detailed Outline............................... vii Acknowledgements............................. xix PART ONE: The New Formal Linguistics Chapter 01: Rectification of Phrase Structure.................. 3 Chapter 02: Escape from the Old Formal Linguistics............... 14 Chapter 03: The Central Result of the New Formal Linguistics........... 21 Chapter 04: The Relationship between Phrase Structure and User-Friendliness..... 27 Chapter 05: Elementary Applications of the New Formal Linguistics......... 37 Chapter 06: Newly Structured Programming................... 45 PART TWO: Establishment of the Context-Oriented Language for Mathematics Chapter 07: Preliminaries.......................... 55 Chapter 08: Definition of COLM, the Context-Oriented Language for Mathematics.... 63 Chapter 09: Early Tutorial on Context-Oriented Symbolic Manipulation........ 76 Chapter 10: The Substitution Operators..................... 83 Chapter 11: Small-Scale Applications: The Qualified Quantifier........... 93 PART THREE: Fundamentals of Context-Oriented Mathematical Logic Chapter 12: The Syntactics of Well-Formed Mathematical Contexts......... 103 Chapter 13: The Semantics of Well-Formed Mathematical Contexts......... 106 Chapter 14: Ramification by Scale of Expression................. 120 Chapter 15: The Central Correspondence between Contexts and Statements...... 131 Chapter 16: The Relations of Strength and Weakness............... 135 Chapter 17: Middle-Scale Applications: The Quote-and-Prove Rules......... 141 PART FOUR: Development of Rules of Inference for the New System Chapter 18: Negations, Complements, and the Dual Symmetry........... 155 Chapter 19: The Theorem on Monotonicity................... 161 Chapter 20: The Local Calculus Rules..................... 170 Chapter 21: The Copy-Migration Rules: Logical Deduction at a Distance....... 187 Chapter 22: The Look-and-Feel of Formal Deduction in the New System....... 193 Chapter 23: Large-Scale Applications: Well-Structured Relativity.......... 203 v
vi Table of Contents PART FIVE: The Structure of Deduction and Proof in the New System Chapter 24: Overview of Context-Oriented Proof Theory.............. 215 Chapter 25: The First Formalization of Context-Oriented Proof........... 222 Chapter 26: Interpolation Operators, Their Warrants and Phrase Structure....... 224 Chapter 27: The Second Formalization of Context-Oriented Proof.......... 234 Chapter 28: Meta-Proof of Theoretical Soundness and Sufficiency.......... 243 Chapter 29: A Full Example of Context-Oriented Formal Proof in Action....... 248 PART SIX: Epilogue Chapter 30: Epilogue............................ 257 Glossary.................................. 259 Bibliography................................ 264 Index................................... 266
Detailed Outline Front Matter Table of Contents.............................. v Detailed Outline.............................. vii Acknowledgements............................. xix PART ONE: The New Formal Linguistics Chapter 1: Rectification of Phrase Structure.................. 3 01 Beyond Words 3 02 On the Title, Well-Structured Mathematical Logic 4 03 On the Names Old and New Formal Linguistics 5 04 On the Names Context-Oriented and Statement-Oriented Mathematical Logic 5 05 A Full Example of Rectified Phrase Structure 6 06 Second Example of Rectified Phrase Structure 8 07 Review of the Literature 11 08 The Method of Gradual Presentation 10 09 What Lies Ahead 12 10 Departure Point for the Expedition 13 Chapter 2: Escape from the Old Formal Linguistics............... 14 01 The Importance of Structure 14 02 Incompatibility of Nested Structures with User-Friendliness 15 03 Description (not Definition) of the term Phrase 18 04 Different Phrase Structures on the Same Expression 19 05 Various Scales of Phrase Structure 20 06 Yet More Systems of Phrase Structure 20 vii
viii Front Matter Chapter 3: The Central Result of the New Formal Linguistics........... 21 01 The Nature of the Central Result 21 02 The Central Result Stated 21 03 Demonstration of the Central Result 21 04 Crossing into the New Formal Linguistics 23 05 Consequences of the Central Result 24 06 Elementary Analytic Applications of the Central Result 25 07 Synthetic Applications of the Central Result 26 Chapter 4: The Relationship between Phrase Structure and User-Friendliness.... 27 01 Hilbert s Meta-Standard 27 02 Another Example of the Problem 28 03 Implicit Phrase Structures 29 04 Relationship of Phrase Structure to User-Friendliness, Shown by Example 31 05 Completely-Nested Phrasing and Its User-Hostility 31 06 Mostly-Flowing Phrasing and Its User-Friendliness 32 07 Conscious and Subconscious Changes of Phrase Structure 34 08 Random-Accessed Phrasing and Its User-Friendliness 34 09 Comments on the Division between the Ground- and Meta-Languages 35 Chapter 5: Elementary Applications of the New Formal Linguistics......... 37 01 Syntactic Craft and Craftsmanship 37 02 A New Outlook 38 03 Finding the Function Space 38 04 A Simple Example 39 05 A Pair of Pre-Compilers for Changing Phrase Structure 41 06 Nest-and-Flow Syntax from Function Spaces with Multiple Inputs 42 07 An Example of Creating a Nest-and-Flow Language 44 Chapter 6: Newly Structured Programming.................. 45 01 Rectification to the Rescue 45 02 Setting Up the Problem to Be Solved 46 03 Creation of Newly Structured Programming 46 04 Example of Newly Structured Programming in Action 49 05 Comments on the Example 50 06 Expert Employment of the One Remaining Nested Form 50 07 Fluvious Programming Languages 52 08 A Night s Rest in the Highlands of Phrase Structure 52
Detailed Outline ix PART TWO: Establishment of the Context-Oriented Language for Mathematics Chapter 7: Preliminaries.......................... 55 01 Before the Threshold 55 02 Some Overarching Concepts 55 03 Several Specific Conventions and Notations 57 04 The Presumed Equivalence Relation on Expressions 60 05 Rules of Precedence 61 06 Thinking at the Proper Granularity 61 Chapter 8: Definition of COLM, the Context-Oriented Language for Mathematics... 63 01 User-Friendliness from the Inside Out 63 02 Overview of Basic COLM 64 03 Definition of the Operators Given and Required, G and R 64 04 Definition of the Quantifiers, A and E 65 05 The Qualifier Such-That, st, and the Qualified Operators 66 06 Digression on the Paradigm of Hypothesis and Conclusion 67 07 Confirmation of Ideas of Phrase Structure 68 08 The Russian Doll Maneuver 70 09 The Negation Operator, n 70 10 The Comment Operator, Called Mark, M 71 11 The Empty Operator, Called Ita, i 71 12 The Wipe-Out Operators, W, Y, w0, and w1 72 13 Operators of Convenience: and, or, if, and iff in a New Phrase Structure 72 14 The Substitution Operators, U and V 73 15 Comments on the Notation of Punctuation 73 16 Translation Guide for the New System 74 17 Sufficiency of Basic COLM for First-Order Expression 75 Chapter 9: Early Tutorial Context-Oriented Manipulation............ 76 01 The Need for a Tutorial 76 02 A Starter Set of Context-Oriented Rules of Inference 77 03 Throwing the Main Clause of a Statement into a Context 78 04 Atomic Translations to Context-Oriented Form 78 05 Solving our Tutorial Problem 78 06 Comparison and Contrast with Traditional Logic 79 07 Preview of Coming Attractions 81 08 Upward and Outward 82
x Front Matter Chapter 10: The Substitution Operators.................... 83 01 The Place of the Substitution Operators in the Grand Scheme of Things 83 02 Overview of Ideas 83 03 The Definition of the Substitution Operators, Part One: Substantial Meanings 85 04 Simple Substitutions 87 05 The Definition of the Substitution Operators, Part Two: Vacuous Meanings 88 06 Actually Performing the Substitution 89 07 Advantages of the New Substitution Operators 90 08 Technical Discussion on Issues of Substitutability 90 09 Technical Section: How to say Uniqueness in the New Language 91 10 Technical Comments about the Substitution Operators 92 Chapter 11: Small-Scale Applications: The Qualified Quantifier.......... 93 01 Improvement as Harder Than It Looks 93 02 The Qualified Quantifier 93 03 The Qualified Quantifier as an Aid to Repairing Fractured Standard Syntax 94 04 A Multi-Purpose Qualification 96 05 COLM as an Assistance to Parsing Dense Technical Prose 97 06 Comments on the Substitution Operators 97 07 How Well-Structured Mathematical Logic Was Built from the Qualified Quantifier 98
Detailed Outline xi PART THREE: Fundamentals of Context-Oriented Mathematical Logic Chapter 12: The Syntactics of Well-Formed Mathematical Contexts........ 103 01 Definitions of Clauses. Species of Clauses. Interiors of Clauses 103 02 Definition of Well-Formed Mathematical Contexts 104 03 Composition of Contexts with Contexts and with Statements 104 04 Phrase Structure of Contexts 105 Chapter 13: The Semantics of Well-Formed Mathematical Contexts........ 106 01 The Existence of Mathematical Situations 106 02 The Idea of Scale of Expression 107 03 The Semantics of G- and R-Clauses 107 04 The Semantics of A- and E-Clauses 109 05 The Semantics of U- and V-Clauses 110 06 The Semantics of Negation 110 07 Aspects of Contexts 111 08 Overloaded Use of Some Words in Informal Mathematics 112 09 The Semantics of Contexts Formed by Composition 113 10 The Necessity of Context-Oriented Thinking for Actually Doing Mathematics 113 11 Further Comments on the Paradigm of Hypothesis and Conclusion 114 12 Limitations to What is Formalized as a Well-Formed Mathematical Context 115 13 Issues of Semantic Soundness and Adequacy 117 14 Technical Section: Contexts of Higher Complexity 118 Chapter 14: Ramification by Scale of Expression................ 120 01 Peano s Dream 120 02 Ramification by Scale of Expression 120 03 Scales of Expression Again 121 04 Examples of Each of the Three Main Scales of Expression 121 05 Self-Similarity of Both Formally and Informally Written Mathematics 123 06 Seeing the Self-Similarity by Zooming In on a Piece of Syntax 124 07 The One-to-One Correspondence between Nestedness and Scale of Expression 125 08 Ramification of Rules of Inference 126 09 COLM as a Useful Shorthand for Writing Mathematics 126 10 On the Intermingling of Hypotheses and Conclusions 127 11 On the Division between Small- and Middle-Scale Expression 129 12 Ramified Thinking 130
xii Front Matter Chapter 15: The Central Correspondence between Contexts and Statements.... 131 01 A Connection between Two Ways of Thinking 131 02 The Central Correspondence Stated 131 03 Details of the Central Correspondence and Why It Is True 131 04 Consequences of the Central Correspondence 132 05 Technical Section: Proof of the Central Correspondence 134 Chapter 16: The Relations of Strength and Weakness.............. 135 01 The Prevalence of the Relations Hence, Because, and Hence-and-Conversely 135 02 The Relations of Para-Strength and Para-Weakness for Statements 135 03 The Relations of Para-Strength and Para-Weakness for Contexts 136 04 The Algebraic Nature of Contexts 137 05 Turning Contexts On and Off 137 06 Para-Positive, Para-Negative, and Zeroish Contexts 138 07 Non-Strict, or Para-, Strength and Weakness as Universal Defaults 138 08 Fundamental Results Concerning the Composition of Contexts 138 09 Relation between Para-Strengthening and the Processes of Proof 139 Chapter 17: Middle-Scale Applications: The Quote-and-Prove Rules........ 141 01 Attaining Ramifiable Closure for How-to-Prove-It Rules 141 02 The Quote-and-Prove Words 141 03 The Quote-and-Prove Rules Stated 142 04 Initial Comments on the Quote-and-Prove Rules 143 05 The Quote-and-Prove Rules Proved 144 06 Examples of the Quote-and-Prove Rules in Action 145 07 The Quote-and-Prove Rules for Negation 148 08 Miscellaneous Comments on the Quote-and-Prove Rules 149 09 Technical Section: On the Closing Sections of Informal Proofs 149 10 A Time for the Expedition Party to Reflect, Relax, and Rest 151
Detailed Outline xiii PART FOUR: Development of Rules of Inference for the New System Chapter 18: Negations, Complements, and the Dual Symmetry.......... 155 01 Arrival at the Construction Site 155 02 General Comments on Negation and the Dual Symmetry 155 03 The Calculus of Negation in COLM 156 04 Tutorial Examples 158 05 Negation in the New and in Traditional Logic Compared and Contrasted 159 06 Complementary Operators 159 07 Technical Section: Tutorial on Forming Duals 160 Chapter 19: The Theorem on Monotonicity.................. 161 01 A Show Horse and a Work Horse 161 02 The Theorem on Monotonicity, Lofty Form 161 03 The Theorem on Monotonicity, Nuts-and-Bolts Statement-Oriented Version 162 04 The Theorem on Monotonicity, Nuts-and-Bolts Context-Oriented Version 163 05 Miscellaneous Results 165 06 A Tutorial Example 166 07 Digression on Logical Variational Analysis 166 08 The Theorem on Monotonicity and the Application of Calculus Rules 167 09 Calculability of Monotonicity 168 10 Comments on the Weakness of Absolute Truth 168 11 Comments on the Strength of Absolute Falsehood 168
xiv Front Matter Chapter 20: The Local Calculus Rules.................... 170 01 The Bricks and Mortar of Logical Deduction 170 02 On the Non-Minimalism of the System 171 03 Collapsing Rules 172 04 Rules for Contraposition 173 05 Contraction Rules 174 06 Rules for Bubble-Busting 175 07 Rules Governing the Insertion of G-, R-, A-, and E-Clauses 176 08 Commutativity and Semi-Commutativity of Clauses 177 09 Further Commutativities When Statements Are Independent of Variables 179 10 Pervasions of Quantifiers, Qualified Quantifiers, and Other Operators 180 11 Rules Governing the Substitution Operator 181 12 Seeing the Substitution Operator in Action 183 13 Stray Rules of Inference 184 14 One Last Rule 184 15 Application of the Local Calculus 184 16 Technical Section: Proof of the Relationship between Independence and Commutativity of Clauses 185 Chapter 21: The Copy-Migration Rules: Logical Deduction at a Distance...... 187 01 The Nature of the Long-Distance Calculus Rules 187 02 The Copy Rules 188 03 The Copy-Migration Rules Stated for Hypotheses and Conclusions 189 04 Copy-Migrations of Quantifiers 190 05 The Copy-Erasure Rule 190 06 The Accumulated Context at a Point 191 07 Technical Section: Proof of the Copy-Migration Rules 191
Detailed Outline xv Chapter 22: The Look-and-Feel of Formal Deduction in the New System...... 193 01 A New Method Is Born 193 02 Tutorial on a Collapsing Rule 193 03 The Definition of Collapsible Clauses and Contexts 194 04 Tutorial on Another Collapsing Rule 194 05 An Ad-Hoc Expository Method 195 06 Proving Our First Substantial Tautology 195 07 Tutorial on Getting Results Out of Syntax Bubbles 196 08 Tutorial on Performing Copy-Migrations with Wild Abandon 197 09 Tutorial on Contraposition 198 10 Example: Showing the Associativity of Disjunction 199 11 The Reduction of Provable Implication 200 12 Seeing the Forest after Seeing the Trees 201 Chapter 23: Large-Scale Applications: Well-Structured Relativity........ 203 01 The Paradigmatic Status of Formal Logic 203 02 The Truth of Statements Relative to Contexts 203 03 Relative Deductive Validity 204 04 Validity Relative to Contradictory Contexts 206 05 Meta-Contexts 207 06 Layered Relativities 208 07 On the Scopes of Variables 208 08 On the Scopes of Hypotheses and Conclusions 209 09 The Difference between the Tiers in the Big Chart 209 10 Reformalization of Logical Axioms as Rules of Inference 210 11 Third Resting Place for the Expedition Party 212
xvi Front Matter PART FIVE: The Structure of Deduction and Proof in the New System Chapter 24: Overview of Context-Oriented Proof Theory............ 215 01 Overview of Proof Theory in General 215 02 Statement- and Context-Oriented Formal Proof Compared and Contrasted 216 03 Very Broad Overview of Different Context-Oriented Proof Apparatus 217 04 Discussion of the First Formalization of Formal Proof 217 05 Discussion of the Second Formalization of Formal Proof 218 06 On the Meta-Proofs of Theoretical Soundness and Sufficiency 220 07 On the Status of Positive Normal Form and of Negative Contexts 220 Chapter 25: The First Formalization of Context-Oriented Proof.......... 222 01 Rules of Inference Reviewed 222 02 The First Formalization of Context-Oriented Proof 222 Chapter 26: Interpolation Operators, Their Warrants and Phrase Structure.... 224 01 The Long, Long Roads of Proof 224 02 The Interpolation Operator eq 224 03 The Interpolation Operators lms, gms, lmw, and gmw 225 04 Warrants of Interpolations 226 05 Personal Phrase Structures of Interpolations 227 06 Example: An Interpolation to Do a Proof by Contraposition 228 07 Example: An Interpolation to Do a Proof by Cases 228 08 Example: Interpolations to Insert Interior Deductions 229 09 Example: Interpolations to Specify Variables 230 10 Example: Interpolations to Add Auxiliary Constructions 231 11 Species of Interpolations 232 Chapter 27: The Second Formalization of Context-Oriented Proof......... 234 01 The Second Formalization Shown by Example 234 02 Positive Normal Form 236 03 Formalization of Sequitur 237 04 The Second Formalization of Context-Oriented Proof 239 05 Technical Section: Informal Sequiturs and Their Taxonomy 240 06 Formal Deductions 240 07 The Relation between Formal and Informal Deduction 242
Detailed Outline xvii Chapter 28: Meta-Proof of Theoretical Soundness and Sufficiency......... 243 01 The Inevitable, at Last 243 02 Meta-Statement of the Soundness and Sufficiency of the Second Formalization of Formal Proof 243 03 Technical Section: Review of the Meta-Proof of Soundness 243 04 Technical Section: Meta-Proof of Sufficiency, Part 1: Setting Up the Works 244 05 Technical Section: Meta-Proof of Sufficiency, Part 2: Creating the Context-Oriented Proof from the Statement-Oriented Proof 244 06 Technical Section: Meta-Proof of Sufficiency, Part 3: Marking All the Clauses Sequitur 245 07 Technical Section: Meta-Proof of Sufficiency, Part 4: Meta-Proof Closure 247 08 Grand Conclusion of the Meta-Proof of Theoretical Soundness and Sufficiency 247 Chapter 29: A Full Example of Context-Oriented Formal Proof in Action...... 248 01 Arrival at Our Final Destination 248 02 Euclid s Proof of the Pythagorean Theorem Translated into Context-Oriented Form 248 03 Comments on the Full Example 252 04 Concluding Remarks 253 PART SIX: Epilogue Chapter 30: Epilogue........................... 257 01 Dismissal of the Expedition Party 257 End Material Glossary.................................. 259 The Greek Alphabet 259 Arithmetic Operators 261 Number Systems 259 Other Symbols 261 Symbols of Traditional Logic 260 Terms 261 Symbols of Set Theory 260 Bibliography................................ 264 01 Authorities on Traditional, Statement-Oriented Logic 264 02 Prior Work in Context-Oriented Logic 264 03 Sources Cited in the Text 264 04 Other Books of Interest 265 Index................................... 266
Acknowledgements A grateful mind By owing owes not, but still pays, at once Indebted and discharg d. Milton. 1 The author would like to thank first, foremost, and especially Richard E. Hodel of Duke University for his support of this endeavor. Dr. Hodel has been a friend to these researches from their earliest infancy through to this their published form. Even the reader has cause to thank Dr. Hodel. Many of the chapters of the current book had their content presented to Dr. Hodel at weekly meetings. With Dr. Hodel acting as first student of the new system, the author was able to tell how to present and how not to present the material in a manner that would be clear and understandable to someone first learning to think in the new way. Dr. Hodel also made numerous suggestions for improvement of the text, and he lent his expertise on occasion to some of the finer points of traditional logic. Barry Lobb, currently at Lynchburg College, should be mentioned because he presented certain calculations in flowing forms and encouraged the author to adopt flowing ways of thinking when the author was an undergraduate student at Butler University. Where Dr. Lobb got these ideas we do not know, whether from somewhere else or from some internal genius, but they form a precursor to the methodical treatment of the flowing form to be found in the following pages. Thanks also are due to Francis Marion University, which has supported this work by granting the author a summer research stipend and a sabbatical leave for the purpose of bringing this work to its present state. Keith Devlin offered the author some crucial advice when the research project was stuck and not advancing properly, and Edward Arroyo and Matthew Turner provided useful comments on drafts of the present work. This work would not have occurred had not Gregory F. Lawler taken the author in as a student and supervised his doctoral dissertation in probability. Credit must also be given to the memory of the late Joseph R. Shoenfield, who taught the author traditional logic during his graduate studies at Duke University. 1. Paradise Lost, Book IV, lines 54 56. xix